We study the robustness of topological ground state degeneracy to long-range interactions in quantum many-body systems. We focus on slowly decaying two-body interactions that scale like a power law $1/r^{\alpha }$ where $\alpha$ is smaller than the spatial dimension; such interactions are beyond the reach of known stability theorems which only apply to short-range or rapidly decaying long-range perturbations. Our main result is a computation of the ground state splitting of several toy models, which are variants of the one-dimensional Ising model $H=-{\sum }_i{\sigma }_i^z{\sigma }_{i+1}^z+\lambda {\sum }_{ij}{|i-j|}^{-\alpha }{\sigma }_i^x{\sigma }_j^x$ with $\lambda >0$ and $\alpha <1$. In one variant, the power-law interactions are replaced by all-to-all interactions $\frac{\lambda }{4L^{\alpha }}{\sum }_{ij}{\sigma }_i^x{\sigma }_j^x$, where $L$ is the system size, while the other variant has true power-law interactions but is built out of quantum rotors rather than Ising spins. These models are also closely connected to the Kitaev $p$-wave wire model with power-law density-density interactions. In these examples, we find that the splitting $\delta$ scales like a stretched exponential $\delta \sim exp(-C{L}^{\frac{1+\alpha }{2}})$. Our computations are based on path integral techniques similar to the instanton method introduced by Coleman. We also study another toy model with long-range interactions that can be analyzed without path integral techniques and that shows similar behavior.

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